Average Error: 0.5 → 0.4
Time: 43.8s
Precision: 64
Internal Precision: 320
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[a \cdot 1 + \left(\left(-\frac{1.0}{3.0}\right) \cdot 1 + \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right) \cdot \left(a - \frac{1.0}{3.0}\right)\right)\]

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.5

    \[\left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right) \cdot \left(\frac{\left(real->posit(1)\right)}{\left(\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\left(real->posit(9)\right) \cdot \left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
  2. Using strategy rm

  3. Applied distribute-rgt-in0.5

    \[\leadsto \color{blue}{\frac{\left(\left(real->posit(1)\right) \cdot \left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right)\right)}{\left(\left(\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\left(real->posit(9)\right) \cdot \left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right)\right)}\right)}\right) \cdot rand\right) \cdot \left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right)\right)}}\]
  4. Using strategy rm

  5. Applied sub-neg0.5

    \[\leadsto \frac{\left(\left(real->posit(1)\right) \cdot \color{blue}{\left(\frac{a}{\left(-\left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right)}\right)}\right)}{\left(\left(\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\left(real->posit(9)\right) \cdot \left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right)\right)}\right)}\right) \cdot rand\right) \cdot \left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right)\right)}\]

  6. Applied distribute-rgt-in0.5

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(a \cdot \left(real->posit(1)\right)\right)}{\left(\left(-\left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right) \cdot \left(real->posit(1)\right)\right)}\right)}}{\left(\left(\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\left(real->posit(9)\right) \cdot \left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right)\right)}\right)}\right) \cdot rand\right) \cdot \left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right)\right)}\]

  7. Applied associate-+l+0.4

    \[\leadsto \color{blue}{\frac{\left(a \cdot \left(real->posit(1)\right)\right)}{\left(\frac{\left(\left(-\left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right) \cdot \left(real->posit(1)\right)\right)}{\left(\left(\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\left(real->posit(9)\right) \cdot \left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right)\right)}\right)}\right) \cdot rand\right) \cdot \left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right)\right)}\right)}}\]

  8. Final simplification0.4

    \[\leadsto a \cdot 1 + \left(\left(-\frac{1.0}{3.0}\right) \cdot 1 + \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right) \cdot \left(a - \frac{1.0}{3.0}\right)\right)\]

Reproduce

herbie shell --seed 2019091 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (*.p16 (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0))) (+.p16 (real->posit16 1) (*.p16 (/.p16 (real->posit16 1) (sqrt.p16 (*.p16 (real->posit16 9) (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0)))))) rand))))